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Question #40792

Difficulty: 8/10 | 🟢 Free

The Question

Let $C_e$ be the conic $x^2 + 2exy + y^2 = e^2$ , where $e \in (0,1)$ is a parameter. Let $A(e)$ be the area of the region in the first quadrant bounded by $C_e$ and the coordinate axes. The value of $e$ that maximizes the function $F(e) = \frac{\sqrt{1-e^2}}{e^2} A(e) + \frac{1}{2} \ln(e)$ is:

Options

A.
$\frac{1}{2}$
B.
$\frac{1}{\sqrt{2}}$
C.
$\frac{\sqrt{3}}{2}$
D.
$\frac{1}{\sqrt{3}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

General Conic Definition|Axis of Integration Selection|Leibniz Rule for Differentiation under Integral|Substitution (Change of Variable)

Why This Question Is Tricky

Common Mistakes

This question tests the calculation of the area of a general conic with a cross-term and the strategic avoidance of the Leibniz Rule trap.

The 'Lethal Collision' occurs if a student attempts to maximize the function by setting up the area as an integral (e.g., $A(e) = \int_0^e (-ex + \sqrt{e^2 - x^2(1-e^2)}) dx$ ) and applying differentiation under the integral sign.

Because both the limits and the integrand depend on the eccentricity $e$ , this leads to a highly complex, seemingly non-elementary integral that is virtually impossible to solve under exam conditions.

The correct path requires recognizing the geometric transformation (Concept 292: Axis/Coordinate Selection) to evaluate the area as an elliptic sector first, reducing it to a closed-form expression before differentiating.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	General Conic Definition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free